W. Knodel, New Gossips and Telephones, Discr. Math. 13, p. 95, 1975.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....of disks in di erent clusters is a certain number of rounds, and one round is required to transfer one unit of data between a pair of disks in the same cluster. 1. 1 Relationship to Gossiping and Broadcasting The problems of Gossiping and Broadcasting have been the subject of extensive study [17, 12, 14, 3, 4, 15, 7, 8]. These play an important role in the design of communication protocols in various kinds of networks. The gossip problem is de ned as follows: there are n individuals. Each individual has an item of gossip that they wish to communicate to everyone else. Communication is typically done in rounds, ....

W. Knodel. New gossips and telephones. Discrete Mathematics, 13:95, 1975.

....and Peters [FP94] investigated the structure of minimum time gossip algorithms using a linear cost model. They established lower and upper bounds on the time to gossip when the number of nodes n is even and showed that there is a synchronous minimum time gossip algorithm for every even n. Knodel [Kno75] showed that gossiping in the unit cost model requires dlog 2 (n)e 1 rounds when n is odd. This lower bound on the number of rounds is also valid for the linear cost model in both the synchronous and asynchronous cases. It is also immediate that at least n steps are required because each node ....

W. Knodel. New gossips and telephones. Discrete Mathematics, 13:95, 1975.

....2 n.# log 2 n. M# . This algorithm on hypercubes is interesting when n is a power of 2 since the disjoint paths of the hypercube can be easily emulated on many other topologies [21] We now present a more general hypercube like algorithm for any even number of vertices based on Knodel graphs [18] of dimension 2 n#. We first recall the definition of Knodel graphs with an even number of vertices [11] Definition 1 Given a set of n vertices, label each vertex p i by the pair (# 1, i mod ) 0 1. The Knodel graph of degree 2 n# has edges between every vertex (1, j) ....

W. Knodel. New gossips and telephones. Discrete Mathematics, 13:95, 1975.

....of information. Therefore, the number of simple rounds required to gossip information between n parties cannot be less than dlog 2 ne, where dxe denotes the smallest integer greater than or equal to x (for the gossip problem it can be shown that the lower bound for odd n is dlog 2 ne 1; see [Kn75]) From these results and observations we derive the following bounds for contributory key distribution systems without broadcasting. Theorem 1 (without broadcasts) Let P be a contributory group key distribution system for n parties not using broadcasts. 1. For the total number of messages 1 ....

....(in round i in the direction b i ) thus, every party is involved in exactly one DH exchange per round. Furthermore, all parties share a common key at the end of this protocol because the vectors b 1 ; b d form a basis of the vector space GF (2) This cube pattern is also used in [Kn75] to manage the gossip problem with a minimum number of rounds. Bu90] suggests using parallel classes in more general geometric structures to distribute information between n parties, which might as well serve as a basis for group key distribution protocols. In order to formulate a protocol for ....

Knodel, W., New Gossips and Telephones. Discrete Mathematics, 13, 1975, 95.

....a piece of information. Therefore, the number of simple rounds required to gossip information between n parties cannot be less than dlog2ne, where dxe denotes the smallest integer greater than or equal to x (for the gossip problem it can be shown that the lower bound for odd n is dlog2ne 1; see [Kn75]) From these results and observations we derive the following bounds for contributory key distribution systems without broadcasting. Theorem 1 (without broadcasts) Let P be a contributory group key distribution system for n parties not using broadcasts. 1. For the total number of messages 1 (P) ....

....(in round i in the direction b i ) thus, every party is involved in exactly one DH exchange per round. Furthermore, all parties share a common key at the end of this protocol because the vectors b1 ; bd form a basis of the vector space GF (2) This cube pattern is also used in [Kn75] to manage the gossip problem with a minimum number of rounds. Bu90] suggests using parallel classes in more general geometric structures to distribute information between n parties, which might as well serve as a basis for group key distribution protocols. In order to formulate a protocol for ....

Knodel, W., New Gossips and Telephones. Discrete Mathematics, 13, 1975, 95.

....12 and deg 1 prime power. For MCages it holds m 1 = D = b g 2 c, which is very small. The sizes of (deg; g) Cages are only known for a limited number of values of deg and g [14] Our experiments showed that among the known Cages only the MCages have fairly small values of m. The Knodel graphs [13] can be constructed for any number of nodes. Their degree is blog 2 nc and their diameter at most dlog 2 ne. For n = 2 i with some i 2 N, their diameter is d log 2 n 2 2 e [8] which improves over the hypercube. It has been shown that Knodel graphs of size 2 i 2 with i 1 are edge symmetric ....

W. Knodel. New gossips and telephones. Discrete Mathematics, 13:95, 1975.

....of time necessary to gossip in graph G, or the gossip time of G. First of all, it is obvious that b(G) g(G) 2 Theta b(G) since gossiping requires at least broadcasting, and since gossiping can be seen as broadcasting followed by a gathering of the information of each vertex. Moreover, Knodel [Kno75] has proved that the gossip time in the complete graph Kn is : ffl dlog 2 ne for n even ; ffl dlog 2 ne 1 for n odd. Such a graph, able to gossip in dlog 2 ne for n even (resp. in dlog 2 ne 1 for n odd) is called a gossip graph, and g n will denote its gossip time. As for broadcasting, it is ....

W. Knodel. New gossips and telephones. Discrete Mathematics, 13:95, 1975.

....a;b : Random graph from G a;b . n = a, m = b, LW = dlog 2 ae odd(a) Here odd(n) n mod 2. Bounds for gossiping in the linear cost model are rare. Obviously, on a network with n PUs, every PU must receive h (n 1) packets. Thus, for any schedule, S h (n 1) Because R dlog ne odd(n) [13], the following trivial lower bound holds for all h, and any network: T dlog ne odd(n) h (n 1) 1) 3 Gossiping on Meshes and Tori In a d dimensional mesh the PUs are laid out on a d dimensional grid. Each PU is connected with its at most 2 d neighbors. A torus is a mesh with ....

Knodel, W., `New Gossips and Telephones,' Discrete Mathematics, 13, p. 95, NorthHolland, 1975.

.... la Lib eration, F33405 Talence Cedex ffertin,raspaudg labri.u bordeaux.fr Abstract Knodel graphs of even order n and degree 1 Delta blog 2 (n)c, W Delta;n , are graphs which have been introduced some 25 years ago as the topology underlying a time optimal algorithm for gossiping among n nodes [Kno75] However, they have been formally defined only 5 years ago [FP94] Since then, they have been widely studied as interconnection networks, mainly because of their good properties in terms of broadcasting and gossiping [BHLP97, FR98] In particular, Knodel graphs of order 2 k , and of degree k, ....

....a study of the different embeddings that can exist between any two of these topologies. Keywords : Knodel graphs, broadcasting, gossiping, interconnection networks, hypercubes, recursive circulant graphs, graph embeddings. 1 Introduction Knodel graphs have been originally introduced in 1975 [Kno75] they were graphs that were underlying Knodel s construction of a time optimal algorithm for gossiping among n vertices, with even n. However, the family of Knodel graphs has been formally defined some 20 years later, by Fraigniaud and Peters [FP94] They are regular graphs of even order n and ....

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W. Knodel. New gossips and telephones. Discrete Mathematics, 13:95, 1975.

....network. This task is usually called broadcasting. When each node knows a piece of information that has to be transmitted to every other node, we are speaking about gossiping. We refer to [6, 8, 9] for surveys on broadcasting and gossiping. The Knodel graph was introduced 25 years ago by Knodel [11], as an interconnection network where gossiping can be performed in the minimum time if the 1 port, telephone model is supposed. More precisely, Knodel gave an algorithm in a complete graph on even number of nodes, so that it allows to complete gossiping in the minimum possible time and Fraigniaud ....

Knodel, W., New gossips and telephones, Discrete Mathematics 13 (1975), 95.

....area of network communications and other areas of parallel and distributed computing. A way to tackle problems like gossiping is to find interconnection networks with the minimum resources necessary to gossip in minimum time. This approach is the one we are dealing with in the following. Knodel [Kno75] proved that the time g n to gossip in the complete graph of order n, K n , under the constant time model is : ffl dlog 2 (n)e for even n, and ffl dlog 2 (n)e 1 for odd n. A Gossip Graph will then denote a graph able to gossip in minimum time. However, it is not necessary to consider the ....

W. Knodel. New gossips and telephones. Discrete Mathematics, 13:95, 1975.

....of edges of a mbg. It was conjectured in [2] that B(2 k Gamma 2) k Gamma 1) 2 k Gamma1 Gamma 1) This has been recently proved by Khachatrian and Harutounian [9] and Dinneen, Fellows, and Faber [4] We present here a short proof due to Monien that points out that, in fact, Knodel [10] has constructed the desired graphs in its solution for a gossiping problem. Theorem 1 For any k, there exists a (k Gamma 1) regular graph G with 2 k Gamma 2 vertices and broadcast time b(G) k. Proof: Let G be a bipartite graph with two parts, each of order 2 k Gamma1 Gamma 1. Vertex ....

W. Knodel. New Gossips and telephones. Discrete Mathematics, vol.13 page 95, 1975.

....and Peters [FP94] investigated the structure of minimum time gossip algorithms using a linear cost model. They established lower and upper bounds on the time to gossip when the number of nodes n is even and showed that there is a synchronous minimum time gossip algorithm for every even n. Knodel [Kno75] showed that gossiping in the unit cost model requires dlog 2 (n)e 1 rounds when n is odd. This lower bound on the number of rounds is also valid for the linear cost model in both the synchronous and asynchronous cases. It is also immediate that at least n steps are required because each node ....

W. Knodel. New gossips and telephones. Discrete Mathematics, 13:95, 1975.

....m 2 ; corresponding to a minimum time gossip scheme on G is called a matching sequence for G. Throughout this paper we consider mgg s G with 2 k vertices (2 k mgg s) In [5] some results are presented for graphs having other numbers of vertices. For the case n = 2 k , we know from [2] that Tn = k. Moreover, it is well known and easy to see that in order to achieve this bound, in every round for every vertex, the number of items it knows must double. Consequently, any matching sequence consists of k pairwise disjoint perfect matchings, i.e. En = k Delta 2 k Gamma1 and every ....

....2 k mgg s is to list all equivalence classes for fixed k. In the following sections we show that for k = 1; 2; 3; 4, there is only one such class. Accordingly, we should introduce a standard gossiping on 2 k vertices which is a well known scheme on the k dimensional hypercube Q k (see e.g. [2]) As m 1 ; m k , take the k pairwise disjoint sets each of 2 k Gamma1 pairwise parallel edges in one fixed direction in Q k . This shows that indeed, Q k is 2 k mgg. Remark: By symmetry, any permutation of these perfect matchings yields a matching sequence for gossiping on Q k . ....

W. Knodel, New gossips and telephones, Discrete Math. 13 (1975) 95.

....has received less attention for the gossiping problem, but there are many papers using this approach for the broadcasting problem. See [13] for a survey, and [3] for recent results. In this paper, we will concentrate on the second approach. In an early paper on minimum time gossiping, Knodel [17] proved that the number of rounds of communication necessary to gossip is dlog ne when n is even, and dlog ne 1 when n is odd. 1 He also proved sufficiency by describing gossip algorithms that meet the lower bounds on numbers of rounds. The graphs underlying Knodel s minimum time gossip ....

....last round of u cannot begin before time t 2 = d Gamma 1)fi (max(jM u j; jM v j) Gamma 1) The total time is therefore t 1 t 2 dlog kefi (k Gamma 1) Lemma 2.3 For any fi 0 and any 0, g fi; n) dlog nefi (n Gamma 1) The proof of Lemma 2. 3 is based on Knodel s proof [17] that it is possible to gossip in dlog ne rounds (where each round takes one unit of time independent of the number of pieces of information exchanged) We define the family of graphs underlying Knodel s construction. Definition 1 The Knodel graph on n 2 nodes (n even) and of maximum degree ....

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W. Knodel. New Gossips and Telephones. Discrete Mathematics 13(1975) 95.

....1. Combining these factors gives total cost log 2 (n)ff P j (j)ffi P j c(j)m(j)L = log 2 (n)ff ( 3n 2 Gamma log 2 (n) Gamma 2)ffi ( n log 2 (n) 2 Gamma n 4 )L . We can do better than the simulated hypercube gossiping algorithm by mapping gossip schemes for Knodel graphs [16] to the cycle. When n = 2 p , the Knodel graph K n is a bipartite graph with n nodes labelled 0 through n Gamma 1 such that node 2i is connected to node 2i 2 j 1 Gamma 3 (mod n) j = 0; p Gamma 1. Nodes 2i and 2i 2 j 1 Gamma 3 (mod n) are considered to be neighbours in ....

.... cycle f0; 1; 2; n Gamma 1g which uses the edges of dimensions 0 and 1 (shown with bold edges in Figure 7) 0 2 4 6 8 10 12 14 1 3 5 7 9 11 13 15 Figure 7: The first three dimensions of the Knodel graph K 16 with its Hamilton cycle Gossip algorithms for Knodel graphs are presented in [16] and [10] The algorithms are similar to gossip algorithms for hypercubes in the sense that each round uses a different dimension and the dimensions can be used in any order. The basic idea of the next cut through algorithm is to map one of the Hamilton cycles of K n to C n and then simulate the ....

W. Knodel. New gossips and telephones. Discrete Mathematics, 13:95, 1975.

....3. 2 Full duplex pairwise communication (F1) For the complete graph with an even number N of nodes (and for the hypercube of N nodes) the time to complete gossip under the F1 model is the same as the time required to broadcast from a single node, or dlg Ne; if N is odd, it is dlg Ne 1 [18]. Farley and Proskurowski [11] have carried out an extensive analysis of the gossip problem under the F1 model for rings and 2 dimensional grids, toroidal grids, and Illiac grids. Their results include the following. For the N node ring the minimum time is N=2 (the diameter) if N is even; if N ....

W. Knodel. New gossips and telephones. Discrete Mathematics, 13:95, 1975.

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W. Knodel, New Gossips and Telephones, Discr. Math. 13, p. 95, 1975.

....the edges of the modified Knodel graph behave similarly to the dimensions in hypercubes. Keywords: broadcasting, gossiping, hypercubes. The second author was supported by the Natural Sciences and Engineering Research Council of Canada under Grant No. OGP0001734. In an early paper on gossiping [2], Knodel gave a dlog ne time gossip scheme for n individuals, where n is even. The scheme used only the edges of a particular subgraph of the complete graph the Knodel graph on n vertices. Although these graphs allow minimum time gossip (and broadcast) a slight modification to the construction ....

W. Knodel, New Gossips and Telephones, Discr. Math. 13, p. 95, 1975.

....can be easily accomplished in a hypercube, by using dimension. See [2] for references on these problems. However, hypercubes can only be constructed for n = 2 d vertices. In this paper, we show that a set of graphs, constructable for any even n, which were used implicitly in a proof of Knodel [3] also have dimension although the notion of dimension is somewhat weaker than that of hypercubes. In this paper, we consider modified Knodel graphs on n vertices (for any even n which is not a power of 2) Let KG n denote the modified Knodel graph on n 2 vertices where n is even and not a power ....

....the operations are modulo n 2 or modulo 2. The edges of KG n are [ x; 0) x 2 i ; 1) for all x and all i where 0 i d Gamma 1 (recall d = blog nc) The edges of the form [ x; 0) x 2 i ; 1) compose a perfect matching and are called edges of dimension i. The graphs used by Knodel [3] are essentially the same, except that he also included edges of the form [ x; 0) x; 1) We refer to those graphs as Knodel graphs and to our graphs as modified Knodel graphs . The edges of a d dimensional hypercube can be partitioned into d sets each set corresponding to a particular ....

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W. Knodel, New Gossips and Telephones, Discr. Math. 13, p. 95, 1975.

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W. Knodel. New gossips and telephones. Discrete Mathematics, 13:95, 1975.

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W. Knodel. New gossips and telephones. Discrete Mathematics, 13:95, 1975.

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W. Knodel. New gossips and telephones. Discrete Mathematics, 13:95, 1975.

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W. Knodel. New gossips and telephones. Discrete Mathematics, 13:95, 1975.

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W. Knodel, New gossips and telephones, Discrete Math. 13 (1975) 95.

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